A combined R-matrix eigenstate basis set and finite-differences propagation method for the time-dependent Schr'{od}dinger equation: the one-electron case

摘要

In this work we present the theoretical framework for the solution of thetime-dependent Schr\"{o}dinger equation (TDSE) of atomic and molecular systemsunder strong electromagnetic fields with the configuration space of theelectron's coordinates separated over two regions, that is regions $I$ and$II$. In region $I$ the solution of the TDSE is obtained by an R-matrix basisset representation of the time-dependent wavefunction. In region $II$ a gridrepresentation of the wavefunction is considered and propagation in space andtime is obtained through the finite-differences method. It appears this is thefirst time a combination of basis set and grid methods has been put forward fortackling multi-region time-dependent problems. In both regions, a high-orderexplicit scheme is employed for the time propagation. While, in a purelyhydrogenic system no approximation is involved due to this separation, inmulti-electron systems the validity and the usefulness of the present methodrelies on the basic assumption of R-matrix theory, namely that beyond a certaindistance (encompassing region $I$) a single ejected electron is distinguishablefrom the other electrons of the multi-electron system and evolves there (regionII) effectively as a one-electron system. The method is developed in detail forsingle active electron systems and applied to the exemplar case of the hydrogenatom in an intense laser field.